# how to deconvolute a array ?

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Rabih Sokhen on 8 Feb 2022
Commented: Chris Turnes on 11 Feb 2022
hy guys
i would like to deconvolute a matrix
code:
clear all
clc
a=rand(10,3);
b=rand10,3); %b=conv2(a,c)
%suppose that b is already the convolution of the array "a" with an array "c"
% I would like to deconvulte " b " to re-obtain "a" and "c".
% any idea how to do so?

Chris Turnes on 9 Feb 2022
Deconvolution is equivalent to polynomial division. You can get the polynomial division and its remainder with the deconv function.
rng('default');
% Take two vectors.
a = randn(7,1);
c = randn(10,1);
% Compute their convolution.
b = conv(a, c);
% "Recover" the first by deconvolving c from b:
[ahat,r] = deconv(b, c);
% Check the residual and the remainder polynomial
norm(a-ahat)
ans = 8.1907e-11
r'
ans = 1×16
1.0e+-9 * 0 0 0 0 0 0 0 0.2664 0.2397 -0.1352 0.2409 0.0603 -0.0114 0.0599 -0.0156 -0.0101
However, it's important to note that this is not a least-squares solution to the deconvolution, and if b isn't really the result convolving something with c, you may not get an answer that's particularly close to the least squares result. To get the least squares result, you would construct the Toeplitz system corresponding to the convolution and solve it:
% Add some "noise" to c:
bhat = conv(a, c + 1e-3*randn(size(c)));
% Solve with deconv:
ahat_deconv = deconv(bhat, c);
% Compare convolving the result with c against the vector we started with:
norm(bhat - conv(ahat_deconv, c))
ans = 6.0224e+03
% Solve with least-squares:
T = convmtx(c, length(a));
ahat_ls = T \ bhat;
% Compare convolving the least-squares result with c against the vector we
% started with:
norm(bhat - conv(ahat_ls, c))
ans = 0.0044
There are efficient algorithms to solve the Toeplitz system, though there are not any functions directly in MATLAB to do so.
Chris Turnes on 11 Feb 2022
Your error here is that you're not specifying the right size for the convolution matrix. The size argument is the size of the thing you are convolving with a -- so in this case, the size of b. You've alredy determined this later, so you just need to pass it into the function:
a = randn(5, 7);
c = randn(13,17);
szB = size(c) - size(a) + 1;
b= full(convmtx2(a, szB) \ c(:));
b= reshape(b, szB);